Лекции по функциональному анализу для начинающих специалистов по математической физике. Арсеньев А.А. - 374 стр.

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A
R
1
3 −ctg(θ/2) 7→ E(−ctg(θ/2) , A) = E
un
(θ , Ca(A))
A
ψ
A
kAψk
2
=
Z
2π
0
(ctg(θ/2))
2
d
θ
< ψ , E
un
(θ , Ca(A))ψ > .
ψ ∈ Dom(A) , φ = (A − iid)ψ , Ca(A)φ = (A + iid)ψ.
ψ =
(φ −Ca(A)φ)
2i
, Aψ =
(φ + Ca(A)φ)
2
.
< ψ , E
un
(α , Ca(A))ψ >=
1
4
Z
α
0
|1 −exp(iθ)|
2
d
θ
< φ , E
un
(θ , Ca(A))φ >=
Z
α
0
(sin(θ/2))
2
d
θ
< φ , E
un
(θ , Ca(A))φ >,
kAψk
2
=
1
4
Z
α
0
|1 + exp(iθ)|
2
d
θ
< φ , E
un
(θ , Ca(A))φ >=
Z
2π
0
(cos(θ/2))
2
d
θ
< φ , E
un
(θ , Ca(A))φ > .
α
θ
1
, θ
2
∈ [ , 2π −] , > 0,
< φ , (E
un
(θ
2
, Ca(A)) −E
un
(θ
1
, Ca(A))φ >=
(sin(θ/2))
−2
(1 + o(1)) < ψ , (E
un
(θ
2
, Ca(A)) −E
un
(θ
1
, Ca(A))ψ >,
   Èç òåîðåìû 4.9.1 ñëåäóåò ÷òî äëÿ îãðàíè÷åííûõ îïåðàòîðîâ A ôóíê-
öèÿ

        R1 3 − ctg(θ/2) 7→ E(− ctg(θ/2) , A) = Eun (θ , Ca(A))              (4.237)

ñîâïàäàåò ñ ââåäåííîé â îïðåäåëåíèè 4.5.3 ñïåêòðàëüíîé ôóíêöèåé, äëÿ
íåîãðàíè÷åííûõ îïåðàòîðîâ A ìû îïðåäåëèì ñïåêòðàëüíóþ ôóíêöèþ
ðàâåíñòâîì (4.237).
Òåîðåìà 4.9.2. Âåêòîð ψ ïðèíàäëåæèò îáëàñòè îïðåäåëåíÿ îïåðàòîðà
A   â òîì è òîëüêî òîì ñëó÷àå, åñëè ñõîäèòñÿ ïîíèìàåìûé êàê íåñîá-
ñòâåííûé èíòåãðàë Ðèìàíà-Ñòèëüòüåñà â ïðàâîé ÷àñòè                      (4.238) è â
ýòîì ñëó÷àå
                   Z   2π
        kAψk = 2
                            (ctg(θ/2))2 dθ < ψ , Eun (θ , Ca(A))ψ > .       (4.238)
                   0

    Äîêàçàòåëüñòâî. Ïóñòü

            ψ ∈ Dom(A) , φ = (A − iid)ψ , Ca(A)φ = (A + iid)ψ.

Òîãäà
                            (φ − Ca(A)φ)        (φ + Ca(A)φ)
                   ψ=                    , Aψ =              .
                                 2i                   2
Ñëåäîâàòåëüíî,

           < ψ , Eun (α , Ca(A))ψ >=
          1 α
            Z
                |1 − exp(iθ)|2 dθ < φ , Eun (θ , Ca(A))φ >=
          4 0
          Z α
              (sin(θ/2))2 dθ < φ , Eun (θ , Ca(A))φ >,                 (4.239)
            0
                    1 α
                      Z
                2
          kAψk =           |1 + exp(iθ)|2 dθ < φ , Eun (θ , Ca(A))φ >=
                    4 0
          Z 2π
               (cos(θ/2))2 dθ < φ , Eun (θ , Ca(A))φ > .               (4.240)
           0

Çàìåòèì, ÷òî ïðàâàÿ ÷àñòü (4.239), âîîáùå ãîâîðÿ, íå äèôôåðåíöèðóåìà
ïî α. Îäíàêî åñëè
                      θ1 , θ2 ∈ [ , 2π − ] ,  > 0,
òî â ñèëó ðàâåíñòâà (4.239):

     < φ , (Eun (θ2 , Ca(A)) − Eun (θ1 , Ca(A))φ >=
    (sin(θ/2))−2 (1 + o(1)) < ψ , (Eun (θ2 , Ca(A)) − Eun (θ1 , Ca(A))ψ >,

                                           362

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